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Statement

Notation (p. 1). For a partition $\mathbb E^N=C_1\cup\cdots\cup C_r$, $\mathbb E^N\xrightarrow{r}X$ means that some $C_i$ contains a set congruent to $X$ for every such partition; a crossed arrow means that some partition has no class containing a congruent copy of $X$. A triangle $T$ is read as the set of its three vertices.

Conjecture 11.1.1 (p. 2). For every triangle $T$ that is not equilateral, $\mathbb E^2\xrightarrow{2}T$: in every partition of the plane into two classes, one class contains a congruent copy of $T$.

Conjecture 11.1.2 (p. 2, labeled "stronger", quoted). "For any partition $\mathbb E^2=C_1\cup C_2$, every triangle occurs (up to congruence) in $C_1$, or else the same holds for $C_2$, with the possible exception of a single equilateral triangle."

The chapter shows that the exception cannot be dropped (p. 2): colour the plane in alternating half-open horizontal strips of width $1$, with $C_1$ the points $(x,y)$ having $2m\le y<2m+1$ for some integer $m$ and $C_2=\mathbb E^2\setminus C_1$. No class contains an equilateral triangle of side $\sqrt3$. The chapter adds, as a further conjecture, that apart from the choice of colour on the boundary lines $y=m$ this is the only partition into two classes avoiding some triangle.

Conjecture 11.1.3 (p. 2). For every triangle $T$, $\mathbb E^2$ is not 3-Ramsey for $T$: the print writes $\mathbb E^2$ with a crossed arrow over $3$ to $T$, so some partition of the plane into three classes has no class containing a congruent copy of $T$.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of the Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press (2004), read in the preprint of the chapter identified on the source card, whose own page numbers are cited: the notation on p. 1, the three conjectures and the strip partition on p. 2.

Read depth. Claims checked: the statements were read clause by clause on the page images of the preprint. They are conjectures, and the chapter proves nothing toward them. Nothing here is independently reviewed.

Proof pointer

None; these are open conjectures as the chapter poses them. The positive cases the chapter lists follow on Theorem 11.1.4.

Dependencies

None.

Bears on

• Problem 173: the problem asks whether in every two-colouring of the plane every triangle but at most one has a monochromatic congruent copy. Conjecture 11.1.2 asserts more, that a single class holds every such triangle and that the possible exception is equilateral, and the strip partition shows that one exception does occur. The chapter states these as conjectures only.